Mathematical Methods for Physics and Engineering : A Comprehensive Guide

24.14 EXERCISES

24.14 Prove that, forα>0, the integral
∫∞

0

tsinαt

1+t^2

dt

has the value (π/2) exp(−α).
24.15 Prove that
∫∞

0

cosmx

4 x^4 +5x^2 +1

dx=

π

6

(

4 e−m/^2 −e−m

)

form> 0.

24.16 Show that the principal value of the integral
∫∞

−∞

cos(x/a)

x^2 −a^2

dx

is−(π/a)sin1.
24.17 The following is an alternative (and roundabout!) way of evaluating the Gaussian
integral.

(a) Prove that the integral of [exp(iπz^2 )]cosecπzaround the parallelogram with

corners± 1 / 2 ±Rexp(iπ/4) has the value 2i.

(b) Show that the parts of the contour parallel to the real axis do not contribute

whenR→∞.

(c) Evaluate the integrals along the other two sides by puttingz′=rexp(iπ/4)

and working in terms ofz′+^12 andz′−^12. Hence, by lettingR→∞show

that

∫∞

−∞

e−πr

2

dr=1.

24.18 By applying the residue theorem around a wedge-shaped contour of angle 2π/n,
with one side along the real axis, prove that the integral
∫∞

0

dx

1+xn

,

wherenis real and≥2, has the value (π/n)cosec (π/n).
24.19 Using a suitable cut plane, prove that ifαis real and 0<α<1then
∫∞

0

x−α

1+x

dx

has the valueπcosecπα.
24.20 Show that ∫
∞

0

lnx

x^3 /^4 (1 +x)

dx=−

√

2 π^2.

24.21 By integrating a suitable function around a large semicircle in the upper half-
plane and a small semicircle centred on the origin, determine the value of

I=

∫∞

0

(lnx)^2

1+x^2

dx

and deduce, as a by-product of your calculation, that

∫∞

0

lnx

1+x^2

dx=0.